JOURNALOl• GEOPHYSICALR•.s•.Aacn
VOL. 72, No. 4
FEBaUAaY 15, 1967
Scaling Law of SeismicSpectrum
K•,ii•i
Axi
Department o• Geology and Geophysics
MassachusettsInstitute oi Technology, Cambridge
The dependenceof the amplitude spectrumof seismicwaves on sourcesize is investigated
on the basis of two dislocationmodels of an earthquake source.One of the models (by N.
Haskell) is called the o•8 model, and the other, called the •2 model, is constructedby fitting
an exponentially decaying function to the autocorrelation function of the dislocation velocity.
The number of source parametersis reduced to one by the assumptionof similarity. We
found that the most convenientparameter for our purposeis the magnitude M,, defined for
surfacewaves with period of 20 sec. Spectral density curves are determined for given M,.
Comparisonof the theoretical curves with observationsis made in two different ways. The
observedratios of the spectra of seismicwaves with the same propagation path but from
earthquakesof different sizesare comparedwith the correspondingtheoretical ratios, thereby
eliminating the effect of propagation on the spectrum. The other method is to check the
theory with the empirical relation between different magnitude scales defined for different waves at different periods.The •2 model gives a satisfactoryagreementwith such observations on the assumptionof similarity, but the •a model doesnot. We find, however,some
indicationsof departure from similarity. The efficiencyof seismicradiation seemsto increase
with decreasingmagnitudeif the Gutenberg-Richtermagnitude-energyrelation is valid. The
assumptionof similarity implies a constantstressdrop independentof sourcesize. A preliminary study of Love waves from the Parkfield earthquake of June 28, 1966, shows that the
stress drop at the source of this earthquake is lower than the normal value (around 100 bars)
by about 2 orders of magnitude.
INTRODUCTION
longer-periodwaves are generated.In the early
days of seismologyin Japan, much attention
years to find seismicsourceparameterssuchas was given to the presenceof large long-period
fault length, rupture velocity,and stressdrop motion in P wavesfrom large local earthquakes
at the earthquakesourcefrom the spectrumof [cf. Matuzawa, 1964]. Analysesof seismicwaves
seismicwaves.Except for the geometricparam- by Jones[1938], Honda and Ito [1939], Gt•teneters obtainedfrom fault plane studies,how- berg and Richter [1942], Byerly [1947], Kanai
ever,the magnitudeis the only physicalparam- et al. [1953], Asada [1953], Aki [1956], Kasaeter that specifiesmost earthquakes.A gap hara [1957], Matumoto [1960], and others
exists between the two approachescurrently have shownthat the period of the spectralpeak
used,onebasedon the useof spectrumand the for P waves,S waves, surfacewaves,and even
otheron amplitude.The purposeof the present for codawaves increaseswith earthquake magElaborate
studies have been made in recent
paper is to fill this gap by finding a first
nitude.
The most convincingevidencefor the greater
efficiencyof generating long-period waves by
basisof somedislocation
modelsof earthquake larger earthquakesis probably given by Bercksources.For this purpose we must reduce to hemer [1962]. He comparedseismogramsobone the number of parametersspecifyinga tained at a station from two earthquakesof the
approximation to the relation between seismic
spectrumand magnitudeof earthquakeson the
dislocation.
We shall make this reductionby same epicenter but of different size. We shall
assumingthat large and small earthquakes reproducehis result later.
satisfy a similarity condition.
The relation betweenseismicspectrumand
The magnitudeof an earthquakeis definedas
a logarithm of amplitude of a certain kind of
earthquakemagnitudeis not a new problem. seismicwave recordedby a certain type of
It has been well known that the greater the band-limited seismograph.If there is such a
size of an earthquake, the more efficiently size effect on seismic spectra as mentioned
1217
1218
KEIITI
above,the unit of magnitudeobtainedfrom one
kind of wave recordedby one type of seismograph may not correspondto that obtained
AKI
1
u0 - 4•rbr
cos
20sin•
from another kind of wave recorded on another
ßw
,t -- r- •/cøs
instrument.In fact, Gutenbergand Richter
[1956a] discovered
a discrepancy
betweenthe where a and b are the velocities of P and S
magnitudescalebaseduponshort-period
body waves,respectively.The aboveexpressions
have
wavesand that baseduponlong-periodsurface the foliowingcommonform:
waves.
It will be shownthat the theoretical
scaling V= P(r, O,,i:,,a, b)
law of the seismicspectrumderived from a
dislocationmodel of the earthquakesource
satisfactorilyexplainsthe above-mentioned
observations.
ßw
15 ,t
½
(3)
where c is the appropriatewave velocity.In
terms of the Fourier transform,the aboveform
THEORETICAL
MODELSOF THE EARTHQUAKE
can be written as
SOURCE
U(o•)= P(r, O,'i:',a, b)A(co)
FollowingHaskell [1966], we definea dislocation functionD(•, t) which is the displacement discontinuityacrossa fault plane at a
point • and time t. The fault plane extends
along the /• axis, and D(•, t) is consideredas
the average dislocationover the width w of the
(4)
where
= u(O,at
A(•o)
=
w
e
dt
fault. Taking the starting point of the fault at
the originof the (x, y, z) coordinates,
and the
ß
,t -r--cøsO
• axis along the x axis, we assumethat the
fault endsat • = L and the surrounding
medium is infinite, isotropic,and homogeneous.If the medim is dissipative,the equationcorIntroducingpolar coordinates
(r, O, •,) by the respondhgto (4) will be
relation
U(oo)
= Pit, O,•, a, b,oo,Q(½o)].
A(½o) (6)
x -'r
cosO
y = r sin 0 cos•v
(1)
where Q(•o) is the dissipationcoefficient.The
above expressionshowsthat .wecan isolatethe
propagation
term P((o) whichdoesnot, except
for the directionof fault propagation,
include
the fault motionparameters.Mathematically,
the displacementcomponentsof P and $ waves sucha simpleisolationis not permittedfor an
at long distances,corresponding
to a sourceof arbitrary heterogeneous
medium.Practically,
longitudinalshear fault [Haskell, 1964] for however,this separationof propagationfactor
example,can be written as
from sourcefactormay be permitted,at least
as a goodfirst approximation.
In this paperwe
z = rsin 0sin•v
shall be concernedonly with the sourcefactor
A((o), which can be calculatedfrom the disloca-
U,.-- 4•rbr sin20sin•
0)
'wfo ,t--r- cos
a
V•
tionD (•, t) according
to (5).
For comparisonwith observationswe shall
use seismicwavesobservedat a given station
fromdistantearthquakes
of thesameepicenter,
thesamefocaldepth,andthesamefaultplane
cos 0 cosqo
solution,but of differentmagnitude.The ratio
ßWfo
r- cos
b
of the Fourier transforms of two such seismo-
grams may be directly comparedwith the
SCALING
LAW
OF SEISMIC
theoretical ratio for the source factor A(o•),
becausethe propagation factor P((o) may be
canceledin the observed ratio. This ingenious
method comes from Berckhemer [1962]. His
theoreticalmodel,however,seemsunrealistic,because,from the point of dislocationtheory, his
model impliesthat the amount of dislocationis
constant,independentof the size of the fault.
Following the generalline of approachtaken
by Haskell [1966],we introduce the autocorrela-
SPECTRUM
1219
•(•, •) = •l•.ff_•
b(•,0 ff• B(•,
ße•(•+')-•½•+•) dk &od• dt
'
I ff:•B(k,o•)B(--k--o•
4•-:
ße
lB(k,o.,)l:
4w2
tionfunction•(•/, •) of •(•, t)'
dk &o
ße
dk &o
Comparingthis formula with (9), we get the
(7)
well-known relation
•(k, o•)= lB(k,o•)[
•'
Putting the Fourier transform of •k(•, •') as
•(k, o•),weget
=ff
(14)
Finally, we get from (9), (13), and (14) the
relationbetweenthe amplitudespectraldensity
•)e
-'•+'•"
d•d. (8)IA(•o)]and the Fouriertransformof the autocorrelation
•(,, •) = •
•(k,w)e'
•-'•"• dk (9)
On the other hand, A(w) can be re•tten by
changingthe order of integration and putting
function'
IA(w)["= w•[(w cosO)/c,w]
(15)
Thus,the sourcefactor]A(co)]of the amplitude
spectral density is expressedin terms of the
t' = t --(r -- • cosO)/cin (5) as follows'
autocorrelation
of dislocation
velocity/)(•, t).
We followed Haskell [1966] in deriving the
A(•)
=we
-'"•fff•b(e,
expressions
above. Haskell, however,calculated
the energyspectraldensityfrom the autocorrelation function$(7, •) of dislocationacceleration
.e-•.•,+•.••o••/• dt• d•
(10) /•(•, o.
In the above e•ression the integration fi•ts
•e extended
to i•ty
by putting•(•, t) = 0
for • ( 0 and L ( •. Putt•g the Fourier trans-
(16)
formof •(•, t) asB(k, w),weobtain
The Fouriertransform
$(k, w) of t•s function
s(•,•)=ff• b(e,
t)e
-'"'*'•
dtde
12ff••
•-•
isrelated
to •(k, w)simply
by
(•)
$(•, •) = • f(•, •)
(•7)
Thus, we obt•n
2
Then we have from (10) and (11)
c
' ,w
•(•)1• = • $[(• •o•0)/•,•]
(12)
and
I.a(•)[: = w•'
c , •o
(13)
On the other hand, we get from (7) and (11)
0s)
As shownabove,the amplitudespectraldensity of seis•c wavescan be expressedin te•
of theautocogelation
function
of •(•, t) orthat
of •(•, t). The autocogelationfunction of
•(•, t) canbe detersnedif the absolu•value
of the Fouriertramformof •(•, t) is •ven.
There •e an i•te
number of space-time
f•ctio•
that •ve a commonspectrMdeity
1220
KEIITI
AKI
but have different phases.By specifyingan
autocorrelation,therefore, we are considering
an infinite group of space-timefunctions.The
model basedupon the autocorrelationfunction
//
is different from the deterministic one in this
respectand may be called 'statistical,'as was
doneby Haskell [1966].
Sincethe earthquakeis essentiallya transient
phenomenon,however,the autocorrelationfunc-
-T
//////
x%
0
T
tion introduced here cannot be treated in the
samemanner as the one for the stationarytime
series.The following figureswill schematically
i]lustrate what form may be expectedfor the
autocorrelation
function
for
the
I!
dislocation
processat an earthquake source. Let the dis-
locationstart at • - 0 and propagatealongthe
• axis with a constantvelocityv; then the dislocationat • will be zero for t • •/v and will
take a constantvalue D0(•) for t • T q- •/v.
Figure I showsa schematicpictureof D(•, t) at
a given•. Thecorresponding/)(•,
t) and•(•, t)
-T
_
"
"/
_• T
•,
)
z:
•,
II
II
Ii
Ii
Ii
Fig. 2. Schematic diagram of autocorrelation
are alsoshownin Figure 1. Their autocorrelation functions of dislocation velocity and dislocation
functionsare shownschematicallyin Figure 2. accelerationat a given point • on a fault.
The dashedlinesin thesefiguresare for the case
in which the dislocation takes the form of a
In ourfirst model,weassumethat the temporal
ramp function in time. We now construct two autocorrelationfunction of dislocationvelocity
earthquake source models by fitting'simple decreases
exponentiallywith the lag r, that is
formulas to the two autocorrelation functions.
f_•b(•,t)b(•,
tq-r)at- •oe
-kr'•l(19)
Our secondmodel is the one proposedby
O(S.t)
r'
T
Haskell. I-Ie assumes that the autocorrelation
function of dislocation acceleration takes the
followingform'
•/,,
b (%.t)
=
(20)
We shall assumean identicalspatial correlation f•ction
>t
for both models. The correlation
betweenthe dislocation
velocityat $ and t and
that at $ + v andt' - t + V/v, that is
II
ff.D(e,
t)D(e
+
',,
de
• in, cate the degreeof pendency of fa•t
propagation.The pers•tency•1 decrease•th
ii
Fig. 1. Schematicdiagram of dislocationand its
time derivativesat a given point • on a fault.
thedistance
v between
thetwopoints.FoHo•ng
•askell, we shahadopt the functionalform of
e-• •,• for t•s expression
,and•so for the co•e-
sportingfunction
of •(•, 0.
SCALING
LAW OF SEISMIC
The above temporal and spatial autocorrela- we have
tion functions are expressedin a single form, if
we write
SPECTRUM
1221
%/4krkr•o
k•,kL
= wDoL
(29)
Inserting this into (25), we get
(21) IA½)I
= •oe-•'"'-•'•-"/"
for the first model,and
wDoL
__
1+ co_s
c 0 2 o•2 {lq-(w/k•)2}
(30)
for our first model.
ße-•'•-""
for the second model. Their Fourier
(22)
transfo•s
are
4k•,kL,•o
Since the above function
decreasesproportionallyto oJ-'-for large oz,we
shall call this the 'o>squaremodel.'
On the other hand, the sourcefactor of amplitude spectral density for our secondmodel will
decrease proportionally to o•-• for large oJ.
q•0is equalto L•D dkrk •,•/8, according
to Haskell.
•(•,•) = /• + (• --•/•)•}(•+ •
Inserting this into (24) and (28), we obtain:
(23)
•(•,•) = /•:•+ (•8k•'kL'q•øa'2
--•/•)•}(•+ • •
(•a)
Using (15) and (23), we may obtain the
sourcefactor of amplitude spectral density for
our first model as follows'
wDoL
½ •)(•'-•'•)}
2•02•/2
{i-q-(
co-so
{1-•'(•)
:•}
(31)
We shall call this the 'o>cube model.'
ASSUMPTION OF SIMILARITY
w¾/4k•,kL
•o
The straightforwardway of testing the earthquake sourcemodelsproposedabove is to compare the predicted spectrum directly with the
observed one. For this purpose, however, we
must know about such effects of the propagation medium as dissipationand complexinterferenees on the seismic spectrum for a wide
frequencyrange. Although such knowledgehas
been accumulating,especially for long-period
waves [ef. Press, 1964], it does not yet satisfactorily cover the frequency range required
for the presentstudy.
As mentioned in the preceding section, we
will removethis difficulty by comparingseismic
waves having a common propagational path
but coming from earthquakesof different sizes.
Further, in order to specify an earthquakeby
a singlesourceparameter,'magnitude,'we must
reduce to one the number of parameters appearingin (30) and (31) by assumingthat they
[k•2
+/.COS
0 •)2(,•2
]1/2+ (.1,)2)
1/2
To determinethe value of •o, we put •o = 0 in
(10). Then
(26)
=w
Comparingthe aboveequationwith (25), we get
%/4krkL
•o= w
(27)
If we define an averagedislocationby
Oo= Z
(28)
are related to each other in some manner.
1222
KEIITI
The simplest of such assumptionsmay be
that large and small earthquakes are similar
phenomena.If any two earthquakesare geometrically similar, the fault width w is proportional to the length L. If they are physically
similar, all the nondimensional
productsformed
by the sourceparameterswill be the same.The
averagedislocationDo will be proportionalto L
and, consequently,to w. This implies that if an
earthquakeis a Starr fracture, the pre-existing
stress or strength is constant and independent
of sourcesize [Tsuboi, 1956]. Since the wave
velocity is practically independent of source
and may be consideredconstantfor our present
purpose,all the quantitieshaving the dimension
of velocity must also be constant and independent of source size. Thus, the similarity
assumptionsimply that the rupture velocity v
is a constantand that all the quantitieshaving
the dimensionof time, suchas k•-• and (vkL)-•,
are proportionalto L.
For simplicity, we shall further assumethat
AKI
y(t)-- w
(32)
I
'" 42
where•oois given by the equation
t - --(d$/dco)
....
(33)
If this approximationis valid, the trace amplitude of waves with frequency • read directly
on the recordwill be proportionalto the spec-
tral density]Y(•)I.
The quantityd•/&o" in
(32) is the sum of a propagation term and a
sourceterm. Sincethe propagationterm is proportional to the travel distance, the source
term may be neglectedat long distances.Thus,
we may assume that the trace amplitude of
surface waves with period of 20 sec is equal
to the amplitude spectral density of waveswith
cos 0 -- 0 and that vkL -- k•. A value of k•
that period, except for a factor that is indegreaterthan vk• may be a more realisticchoice, pendent of the sourcesize. The validity of this
becausek• -• is related to the time required for assumptionis confirmedby comparingthe ratio
formation of fracture across the fault width, of traceamplitudes
of Lovewaveswith a cerwhereas (vk•) -• is related to the time required tain period from two aftershocksof the Kern
for propagationof fracture along the length of County earthquakewith the ratio of amplitude
the fault. We shall examine later the case in
spectral densities at that period obtained by
which 10 vk• -- kr. Essentiallythe sameresult the Fourier analysismethod. Both ratios agree
as when vk• -- kr will be obtained,exceptfor well.
the value of k• correspondingto a sourcesize.
Thus, the dependenceof amplitude spectral
SCALING L•w OF SEISMIC SPECTRUM
Under the assumptionsdescribedin the precedingsection,we can expressthe sourcefactor
of amplitude spectraldensityas a function of
L, % and several nondimensionalconstants.
Taking L as a parameter,we shall obtain a
group of curves of spectral density, each of
whichcorresponds
to an earthquakeof a certain
size. In order to find which curve corresponds
to a given earthquakesize,we must have a scale
to measure size. The most convenient scale for
density,IA(•)I, on the magnitudeM• will be
suchthat log IA(o•)]at the periodof 20 secis
equal to M• plus a constant.In other words,
two spectrum curves corresponding to two
earthquakesizesdiffering by M• -- 1.0 will be
separated by 1.0 along the ordinate at the
periodof 20 sec,if the curveIA(•)I is drawn
on a logarithmic scale. Figures 3 and 4 shows
such groups of curves for the o•-squareand
•-cube models,respectively.
The
curves shown in each of these charts
have an identical shape. The frequency that
our purposeis the surfacewavemagnitudescale, characterizesthe shape of the curve, such as
definedby Gutenbergand Richter [1936]. This k•, is proportionalto L -•, and the spectraldenmagnitude,designatedas Ms, is proportionalto sity at • -- k• is proportionalto L ', as can be
the logarithm of amplitude of teleseismicsur- found from (30) and (31) under the assumpface waves with period of about 20 sec. Since tion of similarity. Therefore, the points corat this period the waves are usually well dis- respondingto the characteristicfrequencylie
persed,we may expressthe wave train y(t) by on a straight line with gradient 3, as shownby
the stationaryphaseapproximation,as follows: dashedlines in Figures 3 and 4. As mentioned
SCALING LAW OF SEISMIC SPECTRUM
PERIOD
0.1
0.2
0.5
I
2
tO-SQUARE
IN
5
I0
SEC
20
50
I00 200
nitudes.The magnitudeof earthquakesstudied
by him coversthe range 4.5 to 8. After several
trials, we choosethe absolutevalue of magnitude that gives the best agreement between
theory and observation.The valuesassignedto
the curves in Figures 3 and 4 are determined
in this manner, and the correspondingtheoretical spectral ratios are shown in Figure 5,
together with the observedratios given by
500 I000
MODEL
1223
'
/
Berckhemer.
LOVE WAVES FROM,Two CALIFORNIA SHOCKS
The applicability of the theoretical curves
of spectral densitiesobtained in the preceding
sectionis tested by the use of recordsof Love
1
waves
f-,/
/
from
two
aftershocks
of
the
Kern
7.0 County, California, earthquake of
8.5
1952. The
epicentersof these two earthquakesare within
severalmiles of each other, accordingto Richter
[1955], and they show identical first motion
patterns, according to Bdth and Richter
PERIOD
,
0.5
I
2
õ
03-CUBE
IN
I0 20
SEC
50 I00 200 500 I000
5(300
MODEL
M,defined
_ 8.o
FREQUENCY IN C/S
,
Fig. 3. I)eper•der•ceof amplitude spectra] density of earthquake magnitude M, for the •-square
model.
'"'
/f •--- 7.5
•
.
before,the spacingof curvesfor differentearthquake magnitudesis determinedby the deft-
6.5
nition of M,. The definitionalone,however,
cannotgivethe absolute
valueof magnitude
•,•
'Y' ,.o
corresponding
to eachcurve.
If
we know the absolute value
for one of
the curves,the values for the rest are determinedfrom the definitionof M•. First we
4.5
adopta trial valueof magnitude
for one of
the curvesand assignmagnitudevaluesto
other curves accordingto the definition. Then
we can find the ratio of spectral densitiesfor
two different magnitudesas a function of frequency or period. This ratio is comparedwith
the observedone given by Berckhemer [1962].
2,0
1.0 0.5
02
0.1 0.05
FREQUENCY
0•2
0.010005
IN
OJ:)02
0.0006
C/S
Fig. 4. Dependence.of amplitude spectral den-
letsdatainclude
sixsetsof twoearthquakes
sityonearthquake
magnitude
M, forthee-cube
with the same epicenter but of different mag-
model.
1224
KEIITI
AKI
AI /A2!
ß
8/6,5
IOO
..
AI/A21
80 -
8/7,5
6
ß
/
5
60
4
40
/
!
20'
ß
I
I
I
I
i0
20
30
40
'-m
I
I
I
I
i
•
50 s
IO
20
30
40
50
60
7,5
/
,
,
AI/A2•
' -- T
70 S
AI/A21
7,4/6,5
50
6,5
20
20
I0
I
I
]
IO
20
30
•
•T
,
,
]
,
I0
40
Ai/A2
t
20
30 S
AI/A21
300
3
200
2
6,2
IOO
,
I
5
I
IO
I
15
• =T
20 S
I
5
I
IO
I
15
/
5,7
I
20 S
Fig. 5. Comparisonof theoretical and observedspectral ratio, plotted against period, for
pairs of earthquakeshaving nearly the same epicenterbut different size. Observedvalues are
reproducedfrom Berclchemer[1962]. The numbers shown for each pair are the earthquake
magnitudefor the pair. Solid line denotes•-square model; dashedline denotes•-cube model.
[1958]. The Richter magnitude(M•,; localscale are obtainedby Berckhemer'smethod,in which
for southernCalifornia) of one of them is 6.1, the ratio is obtainedbetweenthe corresponding
and that of the other is 5.8. The difference of
peaks of waves by directly reading amplitudes
0.3 correspondsto the maximum amplitude on the record. As shown in Figure 7, the corratio of 2 on the record of the standard Woodrespondenceof peaks and troughsbetween the
Andersonseismograph.
two earthquakesis excellent,and there is no
The amplituderatiosof Love wavesfrom the difficulty in obtaining such ratios. Figure 8
two earthquakesobservedat Weston, Ottawa, showsthe ratio of the amplitude spectral denand ResoluteBay are shownin Figure 6. They sity obtained by the Fourier analysismethod.
SCALING
o•O
o
,so,u'r.,¾|
ß WESTON
o7
LAW
OF SEISMIC
SPECTRUM
1225
samespectraldensityat short periodsfor the
two earthquakesand doesnot explain the ob-
I.
•'Ms 0.85
servation.
A more general comparisonof the local
magnitudescaleM•, and the surfacewavemag-
_
nitude scaleM, is difficult.The maximumamplitude recordedby the Wood-Andersonseismo-
//
I
•3model
I0
20
PERIOD
IN
$0
SEC
graph would not be directly proportional to
the amplitude spectral density at any fixed
period,becausethe signaldurationand prevailing period may change with the earthquake
sourcesize. The spectral ratio may be nearly
equalto the maximumamplituderatio for such
Fig. 6. Comparisonof theoreticaland observed earthquakeswith small differencein magnitude
spectral ratio for two aftershocksof the Kern as studiedin the presentsection,but the equalCounty, California,earthquakeof 1952.Observed ity cannothold for larger magnitudedifference.
ratios are obtained from trace amplitude.
Further, the empiricalrelationbetweenM•, and
M,, with which the theoreticalrelation is to be
compared,has not yet beenstabilized[Richter,
There is no significantdifferencebetweenthe
results obtainedby the two methods,justify- 1958].
ing the simpleprocedureusedby Berckhemer.
,
The theoretical curves of spectral density
RELATION BETWEEN ms AND M,
ratio for an earthquake pair with magnitudes
On the other hand, the relation between the
M, around 6.0 which best fit the observations
are shown in these figures. It is remarkable magnitude scale mB, defined as the logarithm
that the observedratio is about 7 at the period of amplitude of teleseismicbody waves, and
M, has been well establishedempirically by
of 20 sec; in other words,the differencein M,
between the two earthquakesis 0.85, about 3 Gutenbergand Richter [1965a]. Further, it is
timeslarger than the differencein M•, obtained well known that the usual record of teleseismic
by Richter. Our u-squaremodel explainsthis body waves,obtainedby a standard short-pefact satisfactorily,becauseM•, must have been riod seismographsuch as Benioff's, shows a
measuredon waves with periods of lessthan 1 rather narrow spectral band around I c/s.
sec, and this model predicts a spectral density Therefore, we may correlate the amplitude of
ratio of about 2 at these periods.On the other bodywaveswith the spectraldensityat I c/s.
If our signal is a finite portion of a Gaussian
hand, the u-cube model predicts nearly the
o)-•
o)
b)•./
•1
IE
WBAY
RESOLUTE
IN
S
WESTON
MINUTE•
S
IN
OTTAWA
Fig. 7. Love wavesfrom the Kern County aftershocks(no. 194 above,no. 141 below; num-
bersassigned
by Richter [1955]) recordedat Ottawa,Resolute,and Weston.
1226
KEIITI
o RESOLUTE
BAY
i-
ß WESTON
10
M, for the to-squareand .to-cubemodel. The
shaded area indicatesthe range between the
• Ms----0.85
above-mentioned two extreme cases of the de-
n OTTAWA
pendenceof spectraldensityon signalduration.
o
o
•.
• M•.=
Q3/,
/
o
o
The theoretical curve for the to-cube model does
'
6
,•o4
2
-•'•
• model
•a modeI
....
0
5
I0
PERIOD
15
IN
20
AKI
25
SEC
Fig. 8. Comparison of theoretical and observed
spectral ratio for two aftershocks of the Kern
County, California, earthquake of 1952. Observed
spectral ratios are obtained by Fourier analysis.
noise, the amplitude spectral density will be
proportional to the square root of the signal
duration. On the other hand, if the signal is a
finite portion of a coherent sinusoidaloscillation, the spectral density will be proportional
to the signal duration. We may assumethat
the actual seismic signal has an intermediate
nature betweenthe above two extremes.Then,
we may write the spectral density at I e/s as
not agree with the empirical one given by
Gutenbergand Richter. On the other hand,the
agreementis excellentfor the to-squaremodel,
exceptfor smallermagnitudes.Looking at the
originaldata from whichGutenbergand Richter
derived their empirical formula, we find that
the theoretical curve based on the to-square
model better explainsthe observationsat small
magnitudesthan the empiricalcurve as shown
in Figure 10. This resultstronglysupportsthe
applicabilityof the scalelaw of seismicspectrum derived from the to-squaremodel on the
assumptionof similarity.
RELATIONBETWEENFAULTLENGTHANDM,
FORTHE (o-SQUAREMODEL
In derivingthe scalinglaw of seismicspectrum
we assumed that the characteristic
fre-
quency k• is proportional to L -'. We can check
this assumption
againstgeological
or geodetic
observations
on an earthquakefault of known
magnitudeM,. The valueof k• for a givenM,
is found from the theoretical curves for the
to-squaremodel shownin Figure 3. Then k•-'
follows:
shouldbe proportional
to L, if the assumption
A(1) = c0nstX A= X tø"•z'ø
(34) of similarity holds.Figure 11 showsthe rela-
whereA., is the maximumtrace amplitudeand
t is the signalduration.
me
According
to GutenbergandRichter [1956b], 8
log t is relatedto m• by the empiricalformula
log t = --1.9 -]- 0.4m•
,
•)-
(35)
Insertingthisequationinto (34), we obtain
log A(1) = cons•-]- (1.2-,• 1.4)ms
,
•-cube
7
model
,
squa
•
(36)
From this equation and the charts of spectral
density curves given in Figures 3 and 4, we
can obtain the theoretical relation between m•
6
/
and M, on the basis of the to-squareand to-cube
models.The constantin (36) is determinedin
such a way that m• and M, agree at 6.75, in
accordance with the Gutenberg-Richter empirical formula
mB= 6.75 + 0.6a(M,- 6.75)
(37)
Figure 9 showsthe relationbetweenrns and
5
'ms=0,63Ms+ 2,5
(Gutenberg-Richter.
1956)
,
5
6
7
8
Ms
Fig. 9. Theoretical relation between ms and
M, based upon the •-square and •o-cubemodels,
as comparedwith the Gutenberg-Richterempirical formula.
SCALING
LAW
OF SEISMIC
SPECTRUM
1227
Ms-Ms
f[ [ I I I [ • I I • [ [ I ] i [ [
f,•)- squar e moclel
Ms FROM
SURFACE
WAVES ,
1.21• 0
' [ o•
M•Me =0.4(Ms-7)
.Sr••
.4-
'
A KERN
COUNTY,
195Z
o o
•
-•
o••oo
2-
• •
0-
•
••o•
o
•
-
0 AVERAGES
o•
••••
oo
o
oo
o
o o •o o o
•m•
o
-.4-
o o•
•6-
o
ß•8-
o
[
t
i
I
[
5.0
i
I
I
6.0
I
I
I
I
7.0
I
I
I
8.o
-
I
I
Ms
Fig. 10. Theoretical relation between ms and M, based upon the •-square model, as compared with that observedby Gutenbergand Richter [1965a].
tion betweenTo -- 2,rk•-• and L for the earth•
I0
2.0
To
50
sec
I00
200
500
I000
/
/
5OO
/
o
/
2OO
/
o
o
I00
oo •/ o o
/
,,/'•'"---L: CONST.
XTO
z
20
/
D
o/p/o
6.5 are excluded. In the case of small earth-
quakes,the surfaceevidencemay not revealthe
true fault lengthat the earthquakefocus.There
is alsosuchan ambiguitywith the magnitudeof
smallearthquakesthat the magnitudegiven in
Tocher'slist may or may not be taken as M,.
Consideringthesefacts,we may concludefrom
Figure 11 that geologicaldata do not exclude
the assumptionof similarity.
The characteristictime To for a given M, as
shownin Figure 11 may seema little too large.
It is possibleto reducethis value without affecting significantlythe conclusionsobtained
/
•o
quake fault given in Toeher'slist [Tocher,
1960]. A linear relationshipholdsbetweenL
and To,if earthquakes
smallerthan magnitude
/
above. If we assume that k• -- 10 vk•. instead
of k• = vkL,and if we determinea set of spectral density curves using the data of Berckhemer and others as given above,we find that
o
the value of To becomes about one-third that
o
t
6
6.5
7
I
7.5
8 ---• Ms
given in Figure 11. The agreementbetween
theory and observationis as goodas that shown
in Figures5, 6, and.8, and we find again that
the to-squaremodel explains the relation be-
Fig. 11. Relationbetweenthe lengthof earthquake fault measuredby geologicalor geodetic tween ms and M, and that the to-cube model
means and the characteristic time of the earth-
quake determined from its magnitude on the
basis of the •-square model. A linear relation
between them supportsthe assumptionof similarity.
does not.
EFFICIENCY OF SEISMIC RADIATION
Let us now examinethe efficiencyof seismic
1228
KEIITI
AKI
energy radiation, which must be independent tributed to a difference in the assumed source
of earthquakesourcesizeif the similaritycon- model. As mentioned before, the model of
dition holdsstrictly. We definethe efficiency Berckhemer,
if interpretedby dislocation
theory,
as the ratio of the energyradiatedin the form is the one in which the dislocation is constant
of seismicwavesto the elasticenergyreleased and independentof sourcesize,but the dislocaby the formationof an earthquakefault. If an tion in our modelis proportionalto the linear
earthquakeis a Starr fracture [Starr, 1928],
the fault-released
elasticenergyis proportional
to DolL. Under the assumption
of similarity,
this energywill be proportionalto L 3. If we
knowthe energyfor a certainvalueof Ms, we
can determinethe value for any M8 from the
scalinglaw givenin the preceding
section.Assumingthat log E is 23.7 for M, ---- 7.5 from
the resultof the writer'sstudyon the Niigata
earthquake[Aki, 1966], we get the released
dimension of the source.
DEPARTURE i•ROM SIMILARITY
As mentionedbefore,the assumptionof similarity implies a constant stress drop in all
earthquakes.If the stressdrop differs for two
earthquakes,our scalinglaw will not apply. If
the stressdrop varies systematicallywith respect to such environmental factors as focal
depth, orientation of fault plane, and cruststrain energyfor variousM, as shownin Table mantle structure, we may construct different
1. The energy radiated in the form of seismic scaling laws for different environments. Such
wavesis evaluatedby the Gutenberg-Richter a study of the seismicspectrummay eventually
formula
reveal the distributionof stressdrop or strength
of material in the earth's crust and mantle. For
log E = 11.4 + 1.5M,
(38)
and is also shownin Table I togetherwith its
ratio to the strain energy.The ratio definitely
increases with decreasingmagnitude. Thus,
starting with the assumptionof similarity, we
have endedby denyingit.
It is, however,not impossiblethat a further
refinementof the magnitude-energyrelation
(38) may eventually support the assumption
of similarity with regard to the radiation efficiency, because(38) is based upon several
simplifiedassumptions.
sucha study,however,we shall needmore precise measurements of spectrum over wider
rangesof frequency than are now available, as
well as detailed knowledgeof the propagation
factor of the spectrum.
Even with the present limited knowledgeof
the propagation factor, however, we may
demonstrate
remarkable
differences
in
stress
drops betweensome earthquakesby the use of
long-period surface waves. The earthquakesto
be comparedhere are the Niigata earthquake
of June 16, 1964, and the Parkfield (California)
earthquakeof June28, 1966.
It should be noted here that Bdth and Duda
The stress drop in the Niigata earthquake
[1964], usingBerckhemer's
result [Berckhemer,
was
obtainedby the following procedure[Aki,
1962], reached an entirely different conclusion
on the radiationeffciency.They found that the 1966]. The geometry of fault movement was
efficiencyincreaseswith increasingmagnitude determined from the radiation patterns of P
of the earthquake.This differencemay be at- waves,$ waves[Hirasawa,1966], and G waves.
The spectraldensity of displacementdue to G
waves was estimated for periods of 50 to 200
TABLE 1. ReleasedStrain Energy, Seismic
see, corrected for dissipation and geometric
Wave Energy and Efficiencyof
spreading,and comparedwith the theoretical
Seismic Radiation
excitation function [Haskell, 1964; Ben-Menahem and Harkrider, 1964] correspondingto a
log E,,t
log E,,,*
Ew[E,t
sourceof that geometry.From this comparison
8.5
26.7
24.2
0.003
we estimatedthe•product of rigidity g, area $
8.0
25.2
23.4
0.016
of fault surface, and average dislocationAu,
7.5
23.7
22.7
0.10
which
correspondsto the moment Mo of the
7.0
22.5
21.9
0.25
component couple of the equivalent doublet
6.5
21.6
21.2
0.40
[Maruyama, 1963; Burridgeand Knopof],1964;
* log E,. = 11.4 q- 1.5 M..
Haskell, 1964]. The value of Mo (-- 1• AuS)
SCALING LAW OF SEISMIC
SPECTRUM
1229
for the Niigata earthquake was 3 X 10• dynes Parkfield earthquake. Consideringthat the efcm. All the near field evidence (echo-sound- feet of finite size was significantfor the Niigata
ing survey,aftershockepicenters,and Tsunami earthquake (about a factor of % at a period
source area) indicated a fault length, L, of of 70 see) but probably not for the Parkfield
about 100 kin. The focal depths of the main earthquake,we estimate the ratio of the source
shockand aftershocksindicated a fault width, momentMo for the Parkfield earthquaketo that
for the Niigata earthquake as 1/250. Thus, we
w, of about 20 kin. Assumingthat p -- 3.7 X
10• dynes cm-•, correspondingto a shear ve- get a moment value of about 1 x 10• dynes
locity of 3.6 kin/see and densityof 2.85 g/cm•, for the Parkfieldearthquake.
we obtainedthe value of the averagedislocation
Using the same rigidity value as for the
as 400 cm by inserting the valuesof L, w, and Niigata earthquake and the observedvalues of
p into the equationMo -- 1• Au ß Lw. This fault length and dislocationmentionedbefore,
value agreeswell with thoseobservedby echo- we get a fault width of about 13 km from the
soundingsurveys made just before and after above value of moment. This value of fault
the earthquake[Mogi et al., 1965]. Finally, the width gives us an extremely low estimate of
stress drop was estimated as about 125 bars strain release.Since Knopoff's fracture model
is more appropriate for a strike slip than
with the aid of Starr'stheory [Starr, 1928].
Now, let us comparethe Niigata earthquake Starr's, we estimate the strain releaseby the
with the Parkfield earthquake.The Parkfield formula e -- Au/2w [Knopo#, 1958]. We get
earthquaketook place right on the San Andreas a value of • of 2 X 10-6 and a corresponding
fault
near
Cholame
and Parkfield.
The
PDE
stressdrop of about 0.7 bar, which is indeeda
card of the Coast and GeodeticSurvey reports
the epicenter as (35.9øN, 120.5øW), and the
origin time as 04:26:12.4 GCT, June 28, 1966.
The magnitudeis 5.8, 5.5, and 6• as givenby
the Pasadena,Berkeley, and Palisadesstations,
respectively.Accordingto a personalcommuni-
remarkably low value. Even if there is an order
of magnitudeerror in estimatingthe value of
moment,the stressdrop is still severalbars.
As mentionedbefore, if the stress drop. is
differentbetweentwo earthquakes,the scaling
law derivedin the presentpaper will not apply
cation from Clarence R. Allen and Stewart W.
to them. We found some indication of violation
Smithof the CaliforniaInstitute of Technology, of the scaling law when we compared the
the near field measurements revealed a strike
Parkfieldearthquakewith oneof the aftershocks
slip fault associatedwith this earthquake,its of the Kern Countyearthquake.
length being about 38 km and its offset about
The magnitude of the Parkfield earthquake
5 cm.
given by local stationsis 5.5 (Berkeley) ,• 5.8
G2 waves from this earthquake are clearly (Pasadena). The surface wave magnitude Ms
recordedby long-periodseismographs
at Reso- of this earthquake, calculated from the Love
lute (A = 40ø) and at Ottawa (A = 35ø). The wave amplitude at a period of 20 see recorded
peak-to-peak amplitudes on the records at a at Ottawa, is 6•. This value agrees with the
period of 70 see are a little over 1 mm at both magnitudegivenby the Palisadesstation.
On the other hand, the magnitude of numstations.This correspondsto a spectraldensity
ber 141 aftershock[Richter, 1955] of the Kern
of ground displacementof about 0.04 em seeat
County earthquake is 6.1. Ms for this earththat period.
The G2 waves from the Niigata earthquake quake, calculatedalso from Love wave amplitude at a period of 20 seerecordedat Ottawa,
at, the epicentraldistanceof 35ø to 40ø showa
spectraldensityof about 1.6 em seeat a period is 6.2.
of 70 see for a certain radiation azimuth. If the
Since the variability of seismicamplitudes is
Niigata earthquakesourceis a strike slip fault very large, it is dangerousto draw any conclulike the Parkfield earthquake, and if we ob- sions from measurements at a few stations.
served G waves in the direction of maximum
However, the magnitudevalues above suggest
radiation,we wouldexpecta spectraldensityof that the spectral density for the Parkfield
about 5 em see at a period of 70 see for G2 earthquake may be greater than that for the
waves at A -- 35 ø • 40 ø. This value is about
Kern County aftershock at long periods, and
125 times as large as that observedfrom the smallerat short periods.If so, the two spec-
1230
KEIITI
AKI
trum curvesmust crosseachother, violating the Berckhemer,H., Die Ausdehnungder Bruchfiiche
i'm Erdbebenherd
und ihr Einfiussauf dasseisscalinglaw. This result is expectedif the stress
mische Wellenspektrum, Getlands• Beitr. Geodrop in the Parkfieldearthquakeis lower than
phys.,71, 5-26, 1962.
that in the Kern County aftershock.The reduc- Burridge, R., and L. Knopoff, Body force equivation of stressdrop is equivalentto the reduction
lents. for seismic dislocations, Bull. $eismol.
$oc. Am., 54, 1875-1888,1964.
of Do in (30), and it will shift the spectrum
curves in Figure 3 downward parallel to the Byefly, P., The periodsof local earthquakewaves
ordinate,causingan intersection
with the original curve in the manner described above.
in central California, Bull. $eismol. $oc. Am.,
37, 291-298, 1947.
Gutenberg, B., and C. F. Richter, On seismic
As we haveseenabove,there is a possibility waves, Getlands Beitr. Geophys., 47, 73-131,
that the stressdrop in an earthquakemay vary
1936.
greatly accordingto its geologicalenvironment. Gutenberg,B., and C. F. Richter,Earthquake
magnitude, intensity, energy, and acceleration,
We shall probablyhave to assigndifferentscalBull. $eismol. $oc. Am., 32, 163-191, 1942.
ing lawsto differentenvironments.
This implies Gutenberg, B., and C. F. Richter, Earthquake
that a singleparameter,such as magnitude, magnitude, intensity, energy, and acceleration,
2, Bull $eismol. $oc. Am., 46, 105-145, 1956a.
cannotdescribean earthquakeevenas a rough
measure.The measurementof seismicspectral Gutenberg, B., and C. F. Richter, Magnitude and
energy of earthquakes, Ann. Geof•s. Rome, 9,
density rather than amplitude will becomeincreasingly important. To understand the ob-
1-15, 1956b.
Haskell, N., Total energy and energy spectral
density of elastic wave radiation from propagating faults, Bull. $eismol. $oc. Am., 54, 1811-
servedspectrumin terms of the physicsof the
earthquakesource,however,we shall have to
knowmoreaboutthe effectof the propagation 1842, 1964.
Haskell, N., Total energy and energy spectral
mediaon the spectrumthan we do now.
density of elastic wave radiation from propagating faults, 2, A statistical sourcemodel, Bull.
$eismol. $oc. Am., 56, 125-140, 1966.
lute recordsof the Parkfieldearthquakeavailable Hirasawa, T., Source mechanismof the Niigata
to me.
earthquake of June 16, 1964, as derived from
This researchwas supportedby the Advanced
analysisof body waves, J. Phys. Earth, 14, in
Acknowledgments. I should like to thank Dr.
J. H. Hodgson for making the Ottawa and Reso-
ResearchProjectsAgencyand was monitoredby
the Air Force Office of Scientific Research under
contract AF49 (638)-1632.
•EFERENCES
Aki, K., Corre]ogram
analysisof seismograms
by
meansof a simpleautomaticcomputer,J. •hys.
Earth, 4, 71-79, 1956.
Aki, K., Generationand propagationof G waves
from the Niigata earthquakeof June 16, 1964,
2, Estimation of earthquakemoment, released
energy, and stress-straindrop from the G wave
spectrum,Bull. Earthquake Res. Inst. Tokyo
Univ., 44, 73-88, 1966.
Asada,T., On the relation betweenthe predominant period and maximum amplitude of earth-
quake motions,J. $eismol.$oc. Japan,$er. 2,
6, 69-73, 1953.
B•th, M., and S. J. Duda, Earthquakevolume,
fault plane area, seismicenergy, strain, deformation, and related quantities, Ann. Geofis.
Rome, 17, 353-368, 1964.
Bath, M., and C. F. Richter, Mechanism of the
aftershocksof the Kern County, California,
earthquakeof 1952,Bull. $eismol.$oc. Am., 48,
133-146, 1958.
Ben-Menaham, A., and D. G. Itarkrider, Radiation patterns of seismic surface waves from
buffed dipolar point sources in a fiat stafffled
earth, J. Geophys.Res., 69, 2605-2620,1964.
press,1966.
Honda, H., and H. Ito, On the period of the P
waves and the magnitude of the earthquake,
Geophys.Mag., 13, 155-160, 1939.
Jones, A. E., Empirical studies of some of the
seismic phenomena of Hawaii, Bull. $eismol.
Soc.Am., 28, 313-338, 1938.
Kanai, K., K. Osada,and S. Yoshizawa,The relation between the amplitude and the period
of earthquake motion, Bull. Earthquake Res.
Inst. Tokyo Univ., 31, 45-56, 1953.
Kashara, K., The nature of seismicorigin as inferred from seismologicaland geodetic observations, 1, Bull. Earthquake Res. Inst. Tokyo,
35, 747-532, 1957.
Knopoff, L., Energy releasein earthquakes,Geophys. J., 1, 44-52, 1958.
Maruyama, T., On the force equivalentsof dynamic elastic dislocations with reference to the
earthquake mechanism,Bull. Earthquake Res.
Inst. Tokyo Univ., 41, 467-486, 1963.
Matumoto, T., On the spectral structure of earth-
quakewaves,Bull. EarthquakeRes.Inst. Tokyo,
38, 13-28, 1960.
Matuzawa,T., Study of earthquakes,OhO$hoten,
Tokyo, pp. 45, 207, 1964.
Mogi, A., B. Kawamura, and Y. Iwabuchi,Submarine crustal movement due to the Niigata
earthquake in 1964, in the environs of the Awa
SCALING LAW OF SEISMIC
Shima Island, Japan Sea, J. Geodetic. •oc.
Japan, 10, 180-186, 1965.
Press,F., Long period waves and free oscillations
of the earth, in Researchin Geophysics,vol. 2,
chapter 1, pp. 1-26, The M.I.T. Press, Cambridge, Mass., 1964.
Richter, C. F., Foreshocksand aftershocks,Earthquakesin Kern County, California, during 1952,
Calif Dept. Nat. Resources,Div. Mines, Bull.
SPECTRUM
Starr, A. T., Slip in a crystal and rupture in a
solid due to shear, Proc Cambridge Phil. Soc.,
24, 489-500, 1928.
Tocher, D., Movement on faults, Proc. 2nd World
Conf. Earthquake Engineering, 1, 551-564, 1960.
Tsuboi, C., Earthquake energy, earthquake volume, aftershock area, and strength of the earth's
crust, J. Phys. Earth, 4, 63-66, 1956.
171, 177-198, 1955.
Richter, C. F., Elementary Seismology,p. 347,
W. tI. Freeman and Co., San Francisco,1958.
1231
(Received September 17, 1966.)